Recall, indices \(a,b\)  are 4-dimensional indices; indices \(A,B\)  are two-dimensional indices of a surface parameterized by angles \(\lbrace \theta ,\varphi \rbrace \) . In (REF ) \(\rho \)  is the energy density, \(p\)  is the fluid pressure, \(\sigma _{ab}\)  and \(\Theta \)  are the shear tensor and the expansion of the null geodesics near the event horizon. Also \(\eta \)  is the shear viscosity, \(\zeta \)  is the bulk viscosity and \(\pi ^A\)  is the momentum density. Specially note the presence of the renormalization parameter \(\alpha \) , the role of which is to make the divergent on the true horizon quantities finite (see [1]} and further discussion below for details).
